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Active noise cancellation algorithms for impulsive noise Peng Li a, Xun Yu a,b,n a b

Department of Mechanical & Industrial Engineering, University of Minnesota Duluth, MN 55812, USA Department of Mechanical and Energy Engineering, University of North Texas, Denton, TX 76203, USA

a r t i c l e in f o

abstract

Article history: Received 28 January 2012 Received in revised form 25 September 2012 Accepted 17 October 2012 Available online 21 December 2012

Impulsive noise is an important challenge for the practical implementation of active noise control (ANC) systems. The advantages and disadvantages of popular ﬁltered-X least mean square (FXLMS) ANC algorithm and nonlinear ﬁltered-X least mean M-estimate (FXLMM) algorithm are discussed in this paper. A new modiﬁed FXLMM algorithm is also proposed to achieve better performance in controlling impulsive noise. Computer simulations and experiments are carried out for all three algorithms and the results are presented and analyzed. The results show that the FXLMM and modiﬁed FXLMM algorithms are more robust in suppressing the adverse effect of sudden large amplitude impulses than FXLMS algorithm, and in particular, the proposed modiﬁed FXLMM algorithm can achieve better stability without sacriﬁcing the performance of residual noise when encountering impulses. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Active noise control Impulsive noise Adaptive ﬁlter Nonlinear algorithm FXLMM Stability

1. Introduction Active noise control (ANC) has received considerable research attention in the past decades because of its superior advantages in cancelling low frequency noise than passive methods [1]. ANC has been applied in various industrial applications such as long duct noise cancellation system, car cabin noise cancellation, and active noise reduction headset [1]. Although great progress has been made, there are still some limits on the application of ANC systems, one important challenge is the control of impulsive noises. Impulse noises exist in many real applications, such as stamping machines in manufacturing plants, IV pump sounds in the hospital [2]. Impulsive noise control was ﬁrstly studied in the literature as non-Gaussian stable processes using fractional lower order moments [3,4]. Sun et al. [5] proposed an algorithm that is a slightly modiﬁed version of the FXLMS algorithm by putting ﬁxed thresholds on the reference signal, in order to eliminate the effect of large impulses whose amplitude is above the threshold and prevent the system from becoming unstable. Although this method has advantage of the same computational complexity as that of the FXLMS algorithm, this algorithm may still becomes unstable under highly impulsive circumstance. Akhtar et al. [6,7] modiﬁed and extended Sun’s algorithm by extending the ﬁxed thresholds to the error signal and replacing large valued samples in the reference signal and error signal with appropriate threshold values. Faster convergence speed and better stability were achieved. Qiu et al. studied another approach that minimizes the squared logarithmic transformation of the error signal [8], achieved more robust performance. Bergamasco et al. also added online secondary path estimation for controlling impulsive noises [9]. Recently, FXLMM algorithm, a simple and robust method that employs the least mean M-estimation error objective function, was extensively studied [2,10–13].

n Corresponding author at: University of Minnesota Duluth, Department of Mechanical & Industrial Engineering, 1155 Union Circle, #311098, Denton, TX 76203, USA. Tel.: þ 1 940 565 2742; fax: þ1 940 369 8675. E-mail address: [email protected] (X. Yu).

0888-3270/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2012.10.017

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Fig. 1. Score function of two-part triangular M-estimate function.

In this study, a new modiﬁed FXLMM algorithm is proposed, aiming to control deterministic impulsive noise such as IV pumps in hospital. The performances of the algorithm are investigated both in computer simulations and experiments.

2. Modiﬁed FXLMM algorithm During noise control experiments, FXLMM algorithm still may become unstable when impulse amplitude increases. This is because the step size of the update weight function increases very fast when encountering large amplitude impulses as the adaptive threshold parameters increasing with the residual noise. To overcome this instability, a new function, a two-part skewed triangular M-estimate function FðeÞ, is proposed to replace the Hampel’s three-part redescending M-estimate function of the traditional FXLMM algorithm for step size adaption: 8 0 r jej o x > < e, FðeÞ ¼ ½x 9e9D2 =ðD1 D2 Þsgn ðeÞ, x r jejo D ð1Þ > : 0, jej Z D and the weight update equation is given as: wðn þ 1Þ ¼ wðnÞ þ mp½eðnÞeðnÞx0 ðnÞ

ð2Þ

where p½eðnÞ ¼ FðeÞ=eðnÞ is the weight function. As illustrated in Fig. 1, when jejo x, the weight update process is identical to the FXLMS algorithm. D1 and D2 are the threshold parameters. In the interval ½x, D, the value of the function FðeÞ begins to decrease, following a line between the points ðx, xÞ and (ðD,0Þ, whose slope is greater than FXLMM in this interval, to achieve faster decrease than the FXLMM algorithm and further reduce the effect of large amplitude impulses in the residual noise on the update process of the weight coefﬁcient vector of adaptive ﬁlter wðzÞ, FðeÞ reaches 0 when 9e9 Z D. Both simulation and experiments were conducted using FXLMS, FXLMM and modiﬁed FXLMM algorithm, and the results are analyzed in the following sections.

3. Simulation results Computer simulations were carried out using MATLAB Simulink, using sample frequency of 5000 Hz and a primary noise source of combination of 300, 500 and 700 Hz sinusoidal waves, in which 500 Hz component had amplitude of 0.10 and the other two each had amplitude of 0.05. A 600 Hz sinusoidal impulse of amplitude of 0.3 is added as the impulsive noise. Adaptive ﬁlter length is 32 bits. Step size is chosen as 1 10 5. The threshold parameters were estimated in realtime online, using the method given in [10]. The probabilities yx , yD1 and yD2 for determining the threshold parameters were taken to be 0.05, 0.025 and 0.01, respectively. The residual noises of all three algorithms are shown in Fig. 2. The FXLMM and modiﬁed FXLMM algorithms achieved much better residual noise performance than FXLMS algorithm, in addition to give much better transitional performance. There was no signiﬁcant difference between FXLMM and modiﬁed FXLMM algorithms in residual noise performance, but the proposed modiﬁed FXLMM algorithm gives much better transit converge performance as show in Fig. 2. Different impulse widths (time duration) and amplitudes were tried and similar results were obtained.

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Fig. 2. Comparison of simulated residual noises of FXLMS, FXLMM and modiﬁed FXLMM algorithms in impulses.

Fig. 3. The wave form of impulsive noise.

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4. Experiment results Real-time control experiments were carried out using a dSPACE real time control board. The secondary path identiﬁcation was identiﬁed ofﬂine using LMS algorithm [1]. Sample frequency is 5000 Hz. Adaptive ﬁlter length is 32 bits. Step size is chosen as 1 10 5. The probabilities yx , yD1 and yD2 for determining the threshold parameters were taken to be 0.05, 0.025 and 0.01, respectively. Noise source used in experiments was sinusoid signal (500 Hz, amplitude 0.1) combined with sinusoid impulsive turbulence (600 Hz, amplitude 0.6). Fig. 3 shows the wave form of impulsive noise in the acoustic box used in the experiments. The experimental residual noises at error microphone are shown in Fig. 4. The FXLMM and modiﬁed FXLMM algorithms achieved signiﬁcantly better performance than FXLMS. While no signiﬁcant difference in the residual noise was observed between FXLMM and modiﬁed FXLMM algorithms, the modiﬁed FXLMM algorithm gave smoother converge performance. The wðnÞ updating trajectories of the three algorithms with presence of impulses are shown in Fig. 5. The weight coefﬁcient updating of FXLMS algorithm was seriously affected by presence of impulses and became unstable after two impulses, while FXLMM algorithm can effectively reduced this negative impact; the modiﬁed FXLMM algorithm had even better stability in coefﬁcient updating than FXLMM algorithm (note the different scales in y axis of the three subplots in Fig. 5). Different impulse widths and amplitudes were implemented and similar results were obtained. In the conducted experiments, modiﬁed FXLMM was able to achieve better stability than FXLMM without sacriﬁcing residual noise performance, which is consistent with simulation results.

Fig. 4. Residual noise of FXLMS, FXLMM and modiﬁed FXLMM algorithms.

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Fig. 5. Comparison of ﬁlter coefﬁcient vector wðnÞ with the updating of FXLMS, FXLMM and Modiﬁed FXLMM algorithms when encountering impulses. (a) FXLMS algorithm. (b) FXLMM algorithm and (c) Modiﬁed FXLMM algorithm.

5. Conclusions Different ANC algorithms for the control of impulsive noises were studied in both simulations and experiments, and their performances were compared. The main advantage of FXLMM and modiﬁed FXLMM is that, when sudden large amplitude impulse happens, it can effectively reduce the effect of impulses on the update of weight coefﬁcient wðnÞ and prevent instability. Compared with FXLMM, through using faster degrading step size optimizing function, the modiﬁed FXLMM algorithm presented in this paper can achieve better stability without sacriﬁcing the performance of residual noise when encountering impulses and has potential of further investigation.

Acknowledgements The authors are grateful for the funding support from the National Institutes of Health (NIH, Grant no. R03DC009673). References [1] S.M. Kuo, D.R. Morgan, Active Noise Control Systems Algorithms and DSP Implementations, John Wiley & Sons, New York, USA, 1996. [2] L. Liu, S. Gujjula, P. Thanigai, M. Kuo, Still in womb: intrauterine acoustic embedded active noise control for infant incubators, Adv. Acoust. Vibr. 2008 (2008) 9, http://dx.doi.org/10.1155/2008/495317. Article ID 495317. [3] M. Shao, C.L. Nikias, Signal processing with fractional lower order moments: stable processes and their applications, Proc. IEEE 81 (1993) 986–1010. [4] R. Leahy, Z. Zhou, Y.C. Hsu, Adaptive ﬁltering of stable processes for active attenuation of impulsive noise, Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 5 (1995) 2983–2986. [5] X. Sun, S.M. Kuo, M. Guang, Adaptive algorithm for active noise control of impulsive noise, J. Sound Vib. 291 (2006) 516–522. [6] M.T. Akhtar, W. Mitsuhashi, Improving performance of FXLMS algorithm for active noise control of impulsive noise, J. Sound Vib. 327 (2009) 647–656. [7] M.T. Akhtar, W. Mitsuhashi, Improving robustness of ﬁltered-x least mean p-power algorithm for active attenuation of standard symmetricoalpha 4-stable impulsive noise, Elsevier Appl. Acoust. 72 (2011) 688–694.

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